Sensitivity to Initial Conditions

[6155GUASTELL@xxxxxxxxxxxxxxx]

From: Ianik Marcil

>IMPREDICTABILITY: epistemological / ontological: a
>simple result of the chaos theories (cf. Ruelle-Takens theory
>of turbulence) is the sensitive dependence on initial
>conditions or on parameters: a micro- variation on the initial
>conditions or on the value of parameters (not on the
>structure of the system) will have, in short or long term,  a
>macro-effect.  Henri Poincare thought in the 1920s that this
>situation was only the result of our incapacity to measure
>adequatly those variations.  This is epistemological
>impredictability.  But chaos theories shown that EVEN if
>the conditions are perfectly measured and known, the
>micro-variations will have an unpredictable macro-behavior.
>(An important problem is the following: we know that the
>behavior is unpredictable, because we use some apparatus to
>describe the behavior of a "real" system; but this apparatus
>is a product of thought, and it is possible to say that this
>apparatus is chaotic, but that the reality that it tempt to
>represent is not chaotic, so this ontological unpredictability
>may be epistemological; anyway, the product of thought has
>a special characteristic in itself.)

So far so good, but there is a growing opinion that sensitivity to
initial conditions is an overindulged concept with respect to
theory development.

First, any function of the type: x2 = f(x1) is sensitive to the
intial value of x1. Some functions are more dramatic than others,
however. For instance x2 = f(c*x1) where <c> is a bifurcation
effect; initial values of one set of <c,x1> values will iterate
differently from another set.

Second, sensitivity to initial conditions will exist within the
basin of a chaotic attractor as everyone here acknowleges.
A point coming in from OUTSIDE the basin, however,  can begin
with any initial conditions, and still end up within the attractor
basin, where it sticks.

--Stephen Guastello